Question

Little Beau Peep lost her sheep. She couldn’t remember how many there were. She knew she would have $400$ more next year, than the number of sheep she had last year. How many sheep were there?
I. The number of sheep last year was $20%$ more than the year before that and this simple rate of increase continues to be the same for the next $10$ years.
II. The increase is compounded annually.

• A. If the question can be answered with the help of statement I alone.
• B. If the question can be answered with the help of statement II alone.
• C. If both statements I and II are needed to answer the question.
• D. If the question cannot be answered even with the help of both statements.

Hint:

In growth problems, identifying whether the increase is simple or compound is crucial. Simple growth adds a constant amount yearly; compound growth increases by a fixed percentage of the current value.

Answer 1

The Correct Option is C

Let the number of sheep last year be $N$. Let the year before last have $N_0$ sheep. From Statement I: \[ N = N_0 \times 1.20 \] Also, the same $20%$ simple rate continues for the next $10$ years. But the problem mentions that she will have $400$ more next year than she had last year, which would be: \[ \text{Next year sheep} = N + 400 \] If the increase were simple interest type, the yearly increment is constant, but if it is compounded, it increases each year proportionally.

Statement II says the increase is compounded annually, which changes the relationship: \[ \text{Next year sheep} = N \times 1.20 \] Given: \[ N \times 1.20 = N + 400 \] \[ 1.20N - N = 400 \] \[ 0.20N = 400 \] \[ N = 2000 \] Thus, combining the compounding info (Statement II) with the $20%$ rate (Statement I) allows direct calculation.
Neither statement alone suffices: Statement I alone assumes simple increase (contradicting actual case), Statement II alone gives no percentage rate.
Hence both together are needed.